-Tamari lattices via subword complexes Cesar Ceballos ( joint with - PowerPoint PPT Presentation

ν -Tamari lattices via subword complexes Cesar Ceballos ( joint with Arnau Padrol and Camilo Sarmiento) The 78th S´ eminaire Lotharingien de Combinatoire Ottrott, March 28, 2016

In this talk Theorem The ν -Tamari lattice is the dual of a well chosen subword complex. 2 2 2 2 1 (2, 0 ,2,2, 1 , 2 ) 0 2 2 2 (2, 0 ,2,1, 1 , 2 ) 1 1 0 2 2 2 (0, 0 ,2,2, 1 , 2 ) 2 1 2 0 (2, 0 ,1,1, 1 , 2 ) 0 1 1 1 0 2 2 1 2 (0, 0 ,2,1, 1 , 2 ) 1 0 0 1 (1, 0 ,1,1, 1 , 2 ) 1 1 1 0 2 (0, 0 ,1,1, 1 , 2 ) 1 1 1 0 0

In this talk Theorem The ν -Tamari lattice is the dual of a well chosen subword complex. 2 2 2 2 1 (2, 0 ,2,2, 1 , 2 ) 0 2 2 2 (2, 0 ,2,1, 1 , 2 ) 1 1 0 2 2 2 (0, 0 ,2,2, 1 , 2 ) 2 1 2 0 (2, 0 ,1,1, 1 , 2 ) 0 1 1 1 0 2 2 1 2 (0, 0 ,2,1, 1 , 2 ) 1 0 0 1 (1, 0 ,1,1, 1 , 2 ) 1 1 1 0 2 (0, 0 ,1,1, 1 , 2 ) 1 1 1 0 0 The picture actually contains three theorems and one corollary. Please remember the picture!

Tamari lattices The Tamari-lattice: partial order on Catalan objects. Tamari. Mono¨ ıdes pr´ eordonn´ es et chaˆ ınes de Malcev. Doctoral Thesis, Paris 1951. Associahedra, Tamari Lattices and Related Structures. Birkh¨ auser/Springer, 2012.

Tamari lattices The Tamari-lattice is a partial order on Catalan objects. Covering relation: A B C A B C Rotation on binary trees

Tamari lattices The Tamari-lattice is a partial order on Catalan objects. Covering relation: Interchanging operation on Dyck paths

m -Tamari lattices Motivated by trivariate diagonal harmonics, F. Bergeron Introduced the m -Tamari lattice on Fuss-Catalan paths. F. Bergeron–Pr´ eville-Ratelle. Higher trivariate diagonal harmonics via generalized Tamari posets. J. Comb 3(3), 2012.

m -Tamari lattices: nice enumerative properties ◮ Number of elements: Fuss Catalan number 1 � ( m +1) n � mn +1 n � ( m +1) 2 n + m ◮ Number of intervals: m +1 � n ( mn +1) n − 1 Chapoton. Sur le nombre d’intervalles dans les treillis de Tamari. S´ em. Lothar. Combin., 55, 2005/07. (m=1) F. Bergeron–Pr´ eville-Ratelle. Higher trivariate diagonal harmonics via generalized Tamari posets. J. Comb 3(3), 2012. (conjectured) Bousquet-M´ elou–Fusy–Pr´ eville-Ratelle. The number of intervals in the m -Tamari lattices. Electron. J. Combin., 18(2), 2011. (proof) ◮ Number of “decorated” intervals: ( m + 1) n ( mn + 1) n − 2 Bousquet-M´ elou–Chapuy–Pr´ eville-Ratelle. The representation of the symmetric group on m -Tamari intervals. Adv. Math., 2013. Conjecture (F. Bergeron (Haiman for m = 1)) The number of intervals is conjecturally interpreted as the dimension of the alternating component of a space in trivariate diagonal harmonics. Decorated intervals correspond to the entire space.

m -Tamari lattices: nice geometry The 2-Tamari lattice for n = 4 C.–Padrol–Sarmiento, 2016: The Hasse diagram of m -Tamari lattices are the edge graphs of (tropical) polytopal subdivisions of associahedra.

ν -Tamari lattices Pr´ eville-Ratelle–Viennot: Introduced the ν -Tamari lattice on lattice paths weakly above ν . Covering relation: Theorem (Pr´ eville-Ratelle–Viennot) This partial order defines a lattice structure on ν -Dyck paths. Pr´ eville-Ratelle–Viennot. An extension of Tamari lattices. To appear in Trans. AMS.

ν -Tamari lattices Pr´ eville-Ratelle–Viennot: Introduced the ν -Tamari lattice on lattice paths weakly above ν . Covering relation: They also have nice enumerative and geometric properties. Fang–Pr´ eville-Ratelle. The enumeration of generalized Tamari intervals. European Journal of Combinatorics 61, 2017. C.–Padrol–Sarmiento. Geometry of ν -Tamari lattices in types A and B . arXiv:1611.09794, 2016.

First theorem Theorem 1 The Hasse diagram of the ν -Tamari lattice is the facet adjacency graph of a well chosen subword complex . This generalizes a known result by Woo (2004), Pilaud–Pocchiola (2010), Stump (2010), and Stump-Serrano (2010) in the classical case.

Subword complexes W = S n +1 group of permutations of [ n + 1] S = { s 1 , . . . , s n } the set of simple generators s i = ( i i + 1) Q = ( q 1 , . . . , q m ) a word in S π ∈ W

Subword complexes W = S n +1 group of permutations of [ n + 1] S = { s 1 , . . . , s n } the set of simple generators s i = ( i i + 1) Q = ( q 1 , . . . , q m ) a word in S π ∈ W Definition (Knutson–Miller, 2004) The subword complex ∆( Q , π ) is the simplicial complex whose faces ← → subwords P of Q such that Q \ P contains a reduced expression of π Knutson–Miller. Gr¨ obner geometry of Schubert polynomials. Ann. Math., 161(3), ’05 Knutson–Miller. Subword complexes in Coxeter groups. Adv. Math., 184(1), ’04

Subword complexes - Example modify s 3 In type A 2 : W = S 3 , S = { s 1 , s 2 } = { (1 2) , (2 3) }

Subword complexes - Example modify s 3 In type A 2 : W = S 3 , S = { s 1 , s 2 } = { (1 2) , (2 3) } Q = ( s 1 , s 2 , s 1 , s 2 , s 1 ) and π = [3 2 1] q 1 , q 2 , q 3 , q 4 , q 5

Subword complexes - Example modify s 3 In type A 2 : W = S 3 , S = { s 1 , s 2 } = { (1 2) , (2 3) } Q = ( s 1 , s 2 , s 1 , s 2 , s 1 ) and π = [3 2 1] q 1 , q 2 , q 3 , q 4 , q 5 q 2 q 3 q 1 ∆( Q , π ) is isomorphic to q 4 q 5

Subword complexes - Example modify s 3 In type A 2 : W = S 3 , S = { s 1 , s 2 } = { (1 2) , (2 3) } Q = ( , s 1 , s 2 , s 1 ) , and π = [3 2 1] = s 1 s 2 s 1 q 1 , q 2 , , , q 2 q 3 q 1 ∆( Q , π ) is isomorphic to q 4 q 5

Subword complexes - Example modify s 3 In type A 2 : W = S 3 , S = { s 1 , s 2 } = { (1 2) , (2 3) } Q = ( s 1 , , s 2 , s 1 ) , and π = [3 2 1] = s 1 s 2 s 1 , q 2 , q 3 , , q 2 q 3 q 1 ∆( Q , π ) is isomorphic to q 4 q 5

Subword complexes - Example modify s 3 In type A 2 : W = S 3 , S = { s 1 , s 2 } = { (1 2) , (2 3) } Q = ( s 1 , s 2 , , s 1 ) , and π = [3 2 1] = s 1 s 2 s 1 , q 3 , q 4 , , q 2 q 3 q 1 ∆( Q , π ) is isomorphic to q 4 q 5

Subword complexes - Example modify s 3 In type A 2 : W = S 3 , S = { s 1 , s 2 } = { (1 2) , (2 3) } Q = ( s 1 , s 2 , s 1 , ) , and π = [3 2 1] = s 1 s 2 s 1 , q 4 , q 5 , , q 2 q 3 q 1 ∆( Q , π ) is isomorphic to q 4 q 5

Subword complexes - Example modify s 3 In type A 2 : W = S 3 , S = { s 1 , s 2 } = { (1 2) , (2 3) } Q = ( , s 2 , s 1 , s 2 , ) and π = [3 2 1] = s 1 s 2 s 1 = s 2 s 1 s 2 q 1 , , q 5 , , q 2 q 3 q 1 ∆( Q , π ) is isomorphic to q 4 q 5

Subword complexes - Example modify s 3 In type A 2 : W = S 3 , S = { s 1 , s 2 } = { (1 2) , (2 3) } Q = ( s 1 , s 2 , s 1 , s 2 , s 1 ) and π = [3 2 1] = s 1 s 2 s 1 = s 2 s 1 s 2 q 1 , q 2 , q 3 , q 4 , q 5 q 2 q 3 q 1 ∆( Q , π ) is isomorphic to q 4 q 5

The subword complex result Theorem 1 The Hasse diagram of the ν -Tamari lattice is the facet adjacency graph of a well chosen subword complex ∆( Q ν , π ν ) . 1 4 3 5 2 6 s 1 s 2 s 3 s 4 s 2 s 4 s 3 s 5 s 3 s 4 Q ν = ( s 3 , s 2 , s 1 , s 4 , s 3 , s 2 , s 4 , s 3 , s 5 , s 4 ) π ν = [1 , 4 , 3 , 5 , 2 , 6]

The subword complex result These objects keep showing up in independent places: Serrano–Stump. Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials. Electron. J. Combin., 19(1), 2012. M´ esz´ aros. Root polytopes, triangulations, and the subdivision algebra. I. Trans. Amer. Math. Soc., 363(8), 2011. Escobar–M´ esz´ aros. Subword complexes via triangulations of root polytopes. arXiv:1502.03997. 1 4 3 5 2 6 s 1 s 2 s 3 s 4 s 2 s 3 s 4 s 5 s 3 s 4 They are special but still some what mysterious.

Facets and ν -trees The facets of ∆( Q ν , π ν ) are given by ν -trees. Two facets are adjacent ↔ the trees are related by rotation. s 2 s 4 s 4 rotation s 2 s 3 s 2 s 3 s 3 s 2 s 3 s 2 s 4 = [1 , 4 , 3 , 5 , 2 , 6] s 3 s 2 s 3 s 4 = [1 , 4 , 3 , 5 , 2 , 6]

Facets and ν -trees The facets of ∆( Q ν , π ν ) are given by ν -trees. Two facets are adjacent ↔ the trees are related by rotation. s 2 s 4 s 4 rotation s 2 s 3 s 2 s 3 s 3 s 2 s 3 s 2 s 4 = [1 , 4 , 3 , 5 , 2 , 6] s 3 s 2 s 3 s 4 = [1 , 4 , 3 , 5 , 2 , 6] ν -tree: (Serrano–Stump) Maximal sets of lattice points above ν avoiding north-east increasing chains p , q such that p � q is above ν . (This talk) some “maximal” binary trees fitting above ν .

The rotation lattice of ν -trees Theorem 1 follows from: Theorem 2 The ν -Tamari lattice is isomorphic to the rotation lattice on ν -trees.

The rotation lattice of ν -trees Theorem 1 follows from: Theorem 2 The ν -Tamari lattice is isomorphic to the rotation lattice on ν -trees. 5 6 6 5 Right flushing 3 4 4 3 1 2 2 1 Left flushing

The lattice of ν -bracket vectors The meet and join: very simple on ν -trees.

The lattice of ν -bracket vectors The meet and join: very simple on ν -trees. Theorem 3 The ν -Tamari lattice is isomorphic to the lattice of ν -bracket vectors under componentwise order. 2 2 2 2 1 (2, 0 ,2,2, 1 , 2 ) 0 2 2 2 (2, 0 ,2,1, 1 , 2 ) 1 1 0 2 2 2 (0, 0 ,2,2, 1 , 2 ) 2 1 2 0 (2, 0 ,1,1, 1 , 2 ) 0 1 1 1 0 2 2 1 2 (0, 0 ,2,1, 1 , 2 ) 1 0 0 (1, 0 ,1,1, 1 , 2 ) 1 1 1 1 0 2 (0, 0 ,1,1, 1 , 2 ) 1 1 1 0 0 b ( T ) = read y -coordinates of the nodes in in-order.